Session 8

Elementary Functions

Prove that

  • exp(a+b) = exp(a)*exp(b)
  • log(a) + log(b) = log(a*b)

Hint1: The solution x(t) of dx = x*dt with x(0) = 1 is x(t) = exp(t).  Choose a, b.  See that y(t)=x(t+a) solves dy = y*dt with y(0) = exp(a) and so (by linearity of dx = x*dt) we have y(t) = exp(a)*exp(t). Conclude that exp(a+b) = x(a+b) = y(b) = exp(a)*exp(b).

Hint2: If A = exp(a) and B = exp(b), then a = log(A) and b = log (B) and since exp(a+b) = exp(a)*exp(b) = A*B, we have log(A*B) = a + b = log(A) + log(B).